Duke Mathematical Journal

Fourier transform on high-dimensional unitary groups with applications to random tilings

Alexey Bufetov and Vadim Gorin

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Abstract

A combination of direct and inverse Fourier transforms on the unitary group U(N) identifies normalized characters with probability measures on N-tuples of integers. We develop the N version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with the law of large numbers and the central limit theorem for global behavior of corresponding random N-tuples.

As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon–Okounkov conjecture (which predicts asymptotic Gaussianity and the exact form of the covariance) for a family of non-simply-connected polygons.

Another application is a central limit theorem for the U(N) quantum random walk with random initial data.

Article information

Source
Duke Math. J., Volume 168, Number 13 (2019), 2559-2649.

Dates
Received: 7 January 2018
Revised: 29 January 2019
First available in Project Euclid: 7 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1567821622

Digital Object Identifier
doi:10.1215/00127094-2019-0023

Subjects
Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
asymptotic representation theory noncommutative Fourier transform random tilings

Citation

Bufetov, Alexey; Gorin, Vadim. Fourier transform on high-dimensional unitary groups with applications to random tilings. Duke Math. J. 168 (2019), no. 13, 2559--2649. doi:10.1215/00127094-2019-0023. https://projecteuclid.org/euclid.dmj/1567821622


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